How to calculate the integral of a erf times another function?

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Discussion Overview

The discussion revolves around the calculation of integrals involving the error function (erf) and other functions, specifically focusing on the integral \(\int^{\infty}_0 e^{-x^2} f(x) dx\) and a related expression involving \(V(k,t)\). The scope includes mathematical reasoning and potential techniques for evaluating these integrals.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks how to calculate the integral \(\int^{\infty}_0 e^{-x^2} f(x) dx\), indicating uncertainty about the function \(f(x)\).
  • Another participant notes that the approach depends on the specific form of \(f(x)\).
  • A different participant introduces a related problem involving the expression \(V(k,t)=V(k,0)e^{-k^2t}+ k\int^{t}_{0}C(t')e^{k^2(t'-t)}dt'\) and its implications as \(k\) approaches infinity.
  • One participant suggests that the term \(V(k,0)\) approaches zero rapidly and can be ignored, while also commenting on the behavior of the integrand.
  • A later reply expresses a need for more detailed understanding and mentions the difficulty in expanding the exponential term in the integral.

Areas of Agreement / Disagreement

Participants express differing views on the evaluation techniques for the integrals, with no consensus on a specific method or resolution of the problems presented.

Contextual Notes

Participants have not provided specific forms for \(f(x)\) or \(C(t')\), and there are unresolved details regarding the mathematical steps involved in the integrals.

jollage
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Hi

I'm encountered the calculation of this function \int^{\infty}_0 e^{-x^2} f(x) dx. How to do it? Thanks.
 
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Depends of what f(x) is.
 
SteamKing said:
Depends of what f(x) is.

SteamKing, thanks. I have another problem, which I encountered in the same paper. Maybe you could help me. Thanks in advance.

The author asserts that V(k,t)=V(k,0)e^{-k^2t}+ k\int^{t}_{0}C(t')e^{k^2(t'-t)}dt' implies V(k,t)=C(t)/k + \mathcal{O}(k^{-3}) when t>0 and k\rightarrow \infty. Could you see this?
 
I haven't worked out the details.
It looks like the V(k,0) term -> 0 very fast, so it can be ignored.
The lntegrand of the integral looks as if -> δ(t'-t)/k2.
 
mathman said:
I haven't worked out the details.
It looks like the V(k,0) term -> 0 very fast, so it can be ignored.
The lntegrand of the integral looks as if -> δ(t'-t)/k2.

Mathman, thanks. I agree with you, but I need the detail to understand it.. Since it seems it is impossible to expand the exponential term in the integral, I don't know other techniques to evaluate the integral...
 

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